Congruence lattice problem

In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem. In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem. We denote by Con A the congruence lattice of an algebra A, that is, the lattice of all congruences of A under inclusion. The following is a universal-algebraic triviality. It says that for a congruence, being finitely generated is a lattice-theoretical property. Lemma.A congruence of an algebra A is finitely generated if and only if it is a compact element of Con A. As every congruence of an algebra is the join of the finitely generated congruences below it (e.g., every submodule of a module is the union of all its finitely generated submodules), we obtain the following result, first published by Birkhoff and Frink in 1948. Theorem (Birkhoff and Frink 1948).The congruence lattice Con A of any algebra A is an algebraic lattice. While congruences of lattices lose something in comparison to groups, modules, rings (they cannot be identified with subsets of the universe), they also have a property unique among all the other structures encountered yet. Theorem (Funayama and Nakayama 1942).The congruence lattice of any lattice is distributive. This says that α ∧ (β ∨ γ) = (α ∧ β) ∨ (α ∧ γ), for any congruences α, β, and γ of a given lattice. The analogue of this result fails, for instance, for modules, as A ∩ ( B + C ) ≠ ( A ∩ B ) + ( A ∩ C ) {displaystyle Acap (B+C) eq (Acap B)+(Acap C)} , as a rule, for submodules A, B, C of a given module.